World War I starts earlier and/or goes on longer; 1899-1909.
Europe is devastated by wars and pandemics, and the only people left with any resources are the people in Liechtenstein.
The US remains isolationist.
The building of the Panama Canal is delayed.
Somebody native to Palau goes the the US, gets a degree in business from some Ivy League university and goes back to Palau and enters politics, becomes influential.
Hawaii is not annexed by the US.
Palau starts making economic ties to the rest of the Pacific.
The rulers of Bahrain marry their daughters into the ruling families of the Arabian Peninsula and then annex the peninsula. Then they marry their daughters or granddaughters into the ruling families of the rest of the Mideast.
During the World War, South Africa becomes autonomous, and the leaders of South Africa make alliances with the leaders of North and South Rhodesia to unite the region as a defense against their neighbors. It doesn't quite work so well. After the union of South Africa with Rhodesia, the whole country is about to collapse into civil war. But an unusually liberal, forward-looking person avoids the civil war by overthrowing the government and renaming the country, "Lesotho."
He gets the support of the whites by invoking the threat of German, Belgian and Dutch invaders from outside the borders. He hires a writer to write a series of epic novels purporting to be fact, connecting "Lesotho" with King Solomon's Mines, Atlantis, the Picts, the Celts, and so on. This encourages the whites to support the new government of "Lesotho".
This propaganda is aimed at the whites.
The propaganda aimed at the blacks is different: Giving the country a Native African name gives the native Africans the impression that they will have a say in the new nation/government, so they support the new leader.
Conditions aren't really all that much better, but it's better than the apartheid of OTL, and there isn't slavery in this Lesotho.
The King of Lesotho is just a figurehead, but a very valuable figurehead who
serves as a propaganda tool to encourage the Africans in the rest of Africa to support Lesotho.
Africa is plunged into years of war--1902-1910. Most of the fighting is outside of Lesotho, which manages to remain aloof, and supports various sides in the various conflicts. Eventually peace is attained. There are about 20 countries in Africa, of which Lesotho is the largest and wealthiest. Each of the other countries' governments is controlled by agents of Lesotho.
When you say Panama rules America, do you mean all of North and Central America? All of North, Central and South America?
Maybe the South won the Civil War but the terms imposed on the South were much harsher, there was a contested Presidential election in 1872 which resulted in further civil war and secessions, and ex-patriates flee to Central America to start their own democracy.
During the period after the US Civil War, to 1915, ex-patriates from the United States, Europe and Africa emigrate to Panama and work on building the Panama Canal, 1905-1915. They are involved in the overthrow of the government of Colombia, and they declare the Republic of Panama, which annexes Colombia. They use bribery, blackmail, extortion and thuggery to take over, intimidate or control the rest of South America.
So Panama is nominally the United States of Panama, but it ends up being a constitutional monarchy. The president is always from one family, the vice-president is always from another family, etc. But there are free elections; citizens can vote on which family member they want for president, vice-president, etc.
Congressional districts are like counties; the borders don't change, they're ruled by one family each, but there are free elections to choose which family member will represent a district.
The Senators are also freely elected, don't have to be from a particular family, and aren't bound to one particular region. But senatorial candidates are chosen by the ruling families. You knew there had to be a catch.
It's not as good as OTL US, but way better than Russia or Rome.
Palau is in the Pacific, so doesn't seem all that plausible for ruling Asia. But here's a try.
Palau gets Hawaii, and becomes an international powerhouse of banking, tourism, gambling, etc.
Civil wars break up Russia, China and Japan. The government of Palau sends in agents to introduce the Asians to the delights of Western-style gambling and other vices. The people eat it right up and support whoever the Palauan agents want.
American ex-patriates fleeing the fighting in the US, and unable to get positions of power and wealth in Panama, seek their fortunes in the Palauan empire and offer their services, helping Palau take control of Asia.
Fascism raises its ugly head in Australia and New Zealand and the Palau agents let the fascists do just enough damage that the Palauans can step in, offer a civilized alternative and take over.
Palauan agents traveling west meet Liechtensteinian agents traveling east. They meet in Crimea at a fancy hotel in 1934 to divide Eurasia and make trade agreements, and avoid a war. Agents from Bahrain are there also, to offer their oil in exchange for recognition and support.
Liechtenstein tried to send agents to the Balkans to make nice and try to introduce them to the good things about life in the Liechtensteinian empire, but the Balkans were too disunited, ornery, impossible to control. So the government of Liechtenstein has some experts draw up a boundary line for the Balkans, beyond which Liechtenstein would not try to exert its influence.
And because the Slovakian warlord seemed a bit more powerful than the other Balkan warlords, the region is called Slovakia. It's still a mess, divided into a varying number of factions, but they don't bother anybody else.
For Koror, the city is moved from the original place in the tiny island to ex-Beijing.
There will still be people living in the islands of Palau, so the city isn't actually moved. Beijing is simply renamed and becomes a secondary capital.
This timeline will require advances in communications as well. Tesla and Edison both live a long time and they and their associates make a lot more progress in radio, TV and other technologies. They refuse to do much in the development of weapons.
This has been fun.